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Theorem sprel 45836
Description: An element of the set of all unordered pairs over a given set 𝑉 is a pair of elements of the set 𝑉. (Contributed by AV, 22-Nov-2021.)
Assertion
Ref Expression
sprel (𝑋 ∈ (Pairs‘𝑉) → ∃𝑎𝑉𝑏𝑉 𝑋 = {𝑎, 𝑏})
Distinct variable groups:   𝑉,𝑎,𝑏   𝑋,𝑎,𝑏

Proof of Theorem sprel
Dummy variable 𝑝 is distinct from all other variables.
StepHypRef Expression
1 elfvex 6907 . 2 (𝑋 ∈ (Pairs‘𝑉) → 𝑉 ∈ V)
2 sprvalpw 45832 . . . 4 (𝑉 ∈ V → (Pairs‘𝑉) = {𝑝 ∈ 𝒫 𝑉 ∣ ∃𝑎𝑉𝑏𝑉 𝑝 = {𝑎, 𝑏}})
32eleq2d 2818 . . 3 (𝑉 ∈ V → (𝑋 ∈ (Pairs‘𝑉) ↔ 𝑋 ∈ {𝑝 ∈ 𝒫 𝑉 ∣ ∃𝑎𝑉𝑏𝑉 𝑝 = {𝑎, 𝑏}}))
4 eqeq1 2735 . . . . . 6 (𝑝 = 𝑋 → (𝑝 = {𝑎, 𝑏} ↔ 𝑋 = {𝑎, 𝑏}))
542rexbidv 3218 . . . . 5 (𝑝 = 𝑋 → (∃𝑎𝑉𝑏𝑉 𝑝 = {𝑎, 𝑏} ↔ ∃𝑎𝑉𝑏𝑉 𝑋 = {𝑎, 𝑏}))
65elrab 3670 . . . 4 (𝑋 ∈ {𝑝 ∈ 𝒫 𝑉 ∣ ∃𝑎𝑉𝑏𝑉 𝑝 = {𝑎, 𝑏}} ↔ (𝑋 ∈ 𝒫 𝑉 ∧ ∃𝑎𝑉𝑏𝑉 𝑋 = {𝑎, 𝑏}))
76simprbi 497 . . 3 (𝑋 ∈ {𝑝 ∈ 𝒫 𝑉 ∣ ∃𝑎𝑉𝑏𝑉 𝑝 = {𝑎, 𝑏}} → ∃𝑎𝑉𝑏𝑉 𝑋 = {𝑎, 𝑏})
83, 7syl6bi 252 . 2 (𝑉 ∈ V → (𝑋 ∈ (Pairs‘𝑉) → ∃𝑎𝑉𝑏𝑉 𝑋 = {𝑎, 𝑏}))
91, 8mpcom 38 1 (𝑋 ∈ (Pairs‘𝑉) → ∃𝑎𝑉𝑏𝑉 𝑋 = {𝑎, 𝑏})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1541  wcel 2106  wrex 3069  {crab 3425  Vcvv 3466  𝒫 cpw 4587  {cpr 4615  cfv 6523  Pairscspr 45829
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-rep 5269  ax-sep 5283  ax-nul 5290  ax-pr 5411  ax-un 7699
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ral 3061  df-rex 3070  df-rab 3426  df-v 3468  df-sbc 3765  df-csb 3881  df-dif 3938  df-un 3940  df-in 3942  df-ss 3952  df-nul 4310  df-if 4514  df-pw 4589  df-sn 4614  df-pr 4616  df-op 4620  df-uni 4893  df-iun 4983  df-br 5133  df-opab 5195  df-mpt 5216  df-id 5558  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-iota 6475  df-fun 6525  df-fv 6531  df-spr 45830
This theorem is referenced by:  prssspr  45837  prsprel  45839  reupr  45874
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